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PS3 Solutions

# PS3 Solutions - PS3 Solutions 1 11.2B from the book...

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PS3 Solutions 1. 11.2B from the book: Consider again (as in the previous problem) the production function x = f ( ) = 100 ℓα . a. Derive the firm’s cost function. Answer: First, we need to invert the production function. We thus take x = 100 ℓα and solve for to get Then we multiply this by w to get b. Derive the marginal and average cost functions and determine how their relationship to one another differs depending on α . Answer: The marginal cost function is: The average cost function is: Notice that AC = MC only when x = 0 unless α = 1 in which case MC = AC everywhere. This is because when α = 1, the production process is linear which means the marginal cost is constant. When α < 1, MC > AC everywhere other than when x = 0; and when α > 1, MC < AC everywhere other than when x = 0. This is because the former represents cases where marginal product of labor diminishes throughout while the latter represents cases where the marginal product of labor increases throughout. c. What is the supply function for this firm when α = 0.5 ? What is the firm’s labor demand curve? Answer: In this case MC > AC everywhere so the MC curve is in fact the supply curve. Put differently, we can set p equal to MC and solve for the output supply of When α = 0.5, this reduces to

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Substituting this into the labor function above and setting α = 0.5 in that equation, we then get the labor demand as d. How do your answers change when α = 1.5? Answer: When α > 1, MC < AC throughout which implies there is no supply curve. The reason is that in this case we have increasing marginal product of labor throughout which implies the price-taking firm would wish to produce an infinite quantity of the output.
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PS3 Solutions - PS3 Solutions 1 11.2B from the book...

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